This invention relates to a polishing simulation, and more particularly to a polishing simulation applied for leveling of concave and convex portions formed on a semiconductor substrate.
Chemical mechanical polishing (CMP) which is an offset leveling technique for a surface of a substrate of a semiconductor device has been and is being employed popularly. However, an analysis of a mechanism of a leveling process has just been placed into argument. Particularly in leveling by chemical mechanical polishing, a pattern density dependency is observed so conspicuously that, if evaluation of a leveled shape of a pattern by working is not performed prior to the leveling, it cannot be determined whether or not the pattern can be accepted as a good pattern.
According to a method which has ordinarily been employed for such estimation, a distribution of stress deformation provided to a polishing cloth by an offset shape of a substrate is calculated in accordance with a finite element method, and a polishing rate for each point of the substrate is estimated based on the distribution. One of methods of the type described is disclosed, for example, in Y. Hayashide et al., VMIC Conference (USA), ISMIC-104/95/0464, 1995, and another method is disclosed in H. Ohtani et al., VMIC Conference (USA), ISMIC-104/95/0447, 1995.
However, since the offset shape varies as polishing proceeds, also the stress distribution varies similarly. Therefore, a polishing rate parameter must be varied. According to the method described above, since calculation of the stress distribution must be performed for each step, a very long time is required for the calculation. Accordingly, an analysis of a substrate which has such a complicated uneven (or concave and convex) shape as allows presence of various test element group (TEG) patterns thereon is very difficult.
It is an object of the present invention to provide a polishing simulation by which an analysis of a substrate having a complicated uneven shape can be performed in a reduced time.
In order to attain the object described above, according to the present invention, there is provided a polishing simulation applied for leveling of an uneven surface of a substrate by polishing, comprising the steps of determining a deformation amount of a polishing cloth on the assumption that a deformed shape of the polishing cloth by a convex portion of the substrate is a truncated cone, determining a distribution of a polishing force based on the deformation amount of the polishing cloth, determining a distribution of a polishing amount of the substrate after a fixed interval of time from the distribution of the polishing pressure, determining a distribution of a height of the substrate from the distribution of the polishing amount after the fixed interval of time, and determining an expression for determination of an offset of the substrate from the distribution of the height of the substrate.
In the polishing simulation, since the deformation amount of the polishing cloth is determined on the assumption that the deformed shape of the polishing cloth by a convex portion of the substrate is a truncated cone, also the deformation of the polishing cloth in the direction of the plane of the polishing cloth is taken into consideration, and consequently, the deformation condition of the polishing cloth is proximate to the deformation condition upon actual polishing. Consequently, the deformation amount of the polishing cloth can be determined almost accurately. Accordingly, the distribution of the polishing pressure which is determined based on the deformation amount of the polishing cloth and hence the distribution of the polishing amount after the fixed interval of time which is determined from the distribution of the polishing pressure can be determined almost accurately. Accordingly, by subtracting the polishing amount after the fixed interval time from the initial state, the distribution of the height of the substrate after the fixed interval of time can be determined almost accurately. Further, since an expression for determination of the offset of the substrate is determined, there is no necessity to calculate any transient solution, and consequently, the speed of calculation is raised and the time required for the calculation can be reduced significantly. As a result, the number of available data (number of grids) can be increased to raise the accuracy in calculation so that a highly accurate polishing simulation can be performed in a short time.
The above and other objects, features and advantages of the present invention will become apparent from the following description and the appended claims, taken in conjunction with the accompanying drawings in which like parts or elements denoted by like reference symbols.